Cauchy sequence

shown in blue, as Failed to parse (Missing texvc executable; please see math/README to configure.): n
versus Failed to parse (Missing texvc executable; please see math/README to configure.): x_n.

In mathematics, a Cauchy sequence, named after Augustin Cauchy, is a sequence whose elements become close to each other as the sequence progresses. To be more precise, by dropping enough (but still only a finite number of) terms from the start of the sequence, it is possible to make the maximum of the distances from any of the remaining elements to any other such element smaller than any preassigned positive value.

In other words, suppose a pre-assigned positive real value Failed to parse (Missing texvc executable; please see math/README to configure.): \varepsilon

is chosen.

However small Failed to parse (Missing texvc executable; please see math/README to configure.): \varepsilon

is, starting from a Cauchy sequence and eliminating terms one by one from the start, after a finite number of steps, any pair chosen from the remaining terms will  be within distance Failed to parse (Missing texvc executable; please see math/README to configure.):  \varepsilon 
of each other.

Because Cauchy sequences require the notion of distance, they can only be defined in a metric space. Their utility lies in the fact that in a complete metric space (one where all such sequences are known to converge to a limit), they give a criterion for convergence which depends only on the terms of the sequence itself. This is often exploited in algorithms, both theoretical and applied, where an iterative process can be shown relatively easily to produce a Cauchy sequence, consisting of the iterates.

The notions above are not as unfamiliar as might at first appear. The customary acceptance of the fact that any real number x has a decimal expansion is an implicit acknowledgment that a particular Cauchy sequence of rational numbers (whose terms are the successive truncations of the decimal expansion of x) has the real limit x. In some cases it may be difficult to describe x independently of such a limiting process involving rational numbers.

Generalizations of Cauchy sequences in more abstract uniform spaces exist in the form of Cauchy filter and Cauchy net.

Cauchy sequence of real numbers

A sequence

Failed to parse (Missing texvc executable; please see math/README to configure.): x_1, x_2, x_3, \ldots

of real numbers is called Cauchy, if for every positive real number ε > 0 there is a positive integer N such that for all natural numbers m,n > N

Failed to parse (Missing texvc executable; please see math/README to configure.): |x_m - x_n| < \varepsilon,

where the vertical bars denote the absolute value.

In a similar way one can define Cauchy sequences of complex numbers.

Cauchy sequence in a metric space

To define Cauchy sequences in any metric space, the absolute value Failed to parse (Missing texvc executable; please see math/README to configure.): |x_m - x_n|

is replaced by the distance Failed to parse (Missing texvc executable; please see math/README to configure.): d(x_m, x_n) 
between Failed to parse (Missing texvc executable; please see math/README to configure.): x_m
and Failed to parse (Missing texvc executable; please see math/README to configure.): x_n

.

Formally, given a metric space (M, d), a sequence

Failed to parse (Missing texvc executable; please see math/README to configure.): x_1, x_2, x_3, \ldots

is Cauchy, if for every positive real number ε > 0 there is a positive integer N such that for all natural numbers m,n > N, the distance

Failed to parse (Missing texvc executable; please see math/README to configure.): d(x_m, x_n)

is less than ε. Roughly speaking, the terms of the sequence are getting closer and closer together in a way that suggests that the sequence ought to have a limit in M. Nonetheless, such a limit does not always exist within M.

Completeness

A metric space X in which every Cauchy sequence has a limit (in X) is called complete.

Examples

The real numbers are complete, and one of the standard constructions of the real numbers involves Cauchy sequences of rational numbers.

A rather different type of example is afforded by a metric space X which has the discrete metric ( where any two distinct points are at distance 1 from each other). Any Cauchy sequence of elements of X must be constant beyond some fixed point, and converges to the eventually repeating term.

Counter-example: rational numbers

The rational numbers Q are not complete (for the usual distance): There are sequences of rationals that converge (in R) to irrational numbers; these are Cauchy sequences having no limit in Q. In fact,if a real number x is irrational, then the sequence (x_n), whose n-th term is the truncation to n decimal places of the decimal expansion of x, gives Cauchy sequence of rational numbers with irrational limit x. Irrational numbers certainly exist, for example:

• The sequence defined by x₀ = 1, x_n+1 = (x_n + 2/x_n)/2 consists of rational numbers (1, 3/2, 17/12,...), which is clear from the definition; however it converges to the irrational square root of two, see Babylonian method of computing square root.
• The sequence Failed to parse (Missing texvc executable; please see math/README to configure.): x_n = F_n / F_{n-1}

of ratios of consecutive Fibonacci numbers which, if it converges at all, converges to a limit Failed to parse (Missing texvc executable; please see math/README to configure.): \phi
satisfying Failed to parse (Missing texvc executable; please see math/README to configure.): \phi^2 = \phi+1

, and no rational number has this property. If one considers this as a sequence of real numbers, however, it converges to the real number Failed to parse (Missing texvc executable; please see math/README to configure.): \phi = (1+\sqrt5)/2 , the Golden ratio, which is irrational.

• The values of the exponential, sine and cosine functions, exp(x), sin(x), cos(x), are known to be irrational for any rational value of x≠0, but each can be defined as the limit of a rational Cauchy sequence, using, for instance, the Maclaurin series.

Other properties

• Every convergent sequence (with limit s, say) is a Cauchy sequence, since, given any real number r > 0, beyond some fixed point, every term of sequence is within distance r/2 of s, so any two terms of the sequence are within distance r of each other.
• Every Cauchy sequence of real (or complex) numbers is bounded ( since for some N, all terms of the sequence from the N-th onwards are within distance 1 of each other, and if M is the largest absolute value of the terms up to and including the N-th, then no term of the sequence has absolute value greater than M+1).
• In any metric space, a Cauchy sequence which has a convergent subsequence with limit s is itself convergent (with the same limit), since, given any real number r > 0, beyond some fixed point in the original sequence, every term of the subsequence is within distance r/2 of s, and any two terms of the original sequence are within distance r/2 of each other, so every term of the original sequence is within distance r of s.

These last two properties, together with a lemma used in the proof of the Bolzano-Weierstrass theorem, yield one standard proof of the completeness of the real numbers, closely related to both the Bolzano-Weierstrass theorem and the Heine–Borel theorem. The lemma in question states that every bounded sequence of real numbers has a convergent subsequence. Given this fact, every Cauchy sequence of real numbers is bounded, hence has a convergent subsequence, hence is itself convergent. It should be noted, though, that this proof of the completeness of the real numbers implicitly makes use of the least upper bound axiom. The alternative approach, mentioned above, of constructing the real numbers as the completionhttp://en.wikilib.com/wiki/Completion_%28metric_space%29#Completion of the rational numbers, makes the completeness of the real numbers tautological.

One of the standard illustrations of the advantage of being able to work with Cauchy sequences and make use of completeness is provided by consideration of the summation of an infinite series of real numbers (or, more generally, of elements of any complete normed linear space, or Banach space). Such a series Failed to parse (Missing texvc executable; please see math/README to configure.): \sum_{n=1}^{\infty} x_{n}

is considered to be convergent if and only if the sequence of partial sums Failed to parse (Missing texvc executable; please see math/README to configure.): (s_{m})
is convergent, where 

Failed to parse (Missing texvc executable; please see math/README to configure.): s_{m} = \sum_{n=1}^{m} x_{n} . It is a routine matter to determine whether the sequence of partial sums is Cauchy or not, since for positive integers p > q ,

Failed to parse (Missing texvc executable; please see math/README to configure.): s_{p} - s_{q} = \sum_{n=q+1}^{p} x_{n}

.

If Failed to parse (Missing texvc executable; please see math/README to configure.): f \colon M \rightarrow N

is a uniformly continuous map between the metric spaces M and N and (xn) is a Cauchy sequence in M, then Failed to parse (Missing texvc executable; please see math/README to configure.): (f(x_n))
is a Cauchy sequence in N. If Failed to parse (Missing texvc executable; please see math/README to configure.): (x_n)
and Failed to parse (Missing texvc executable; please see math/README to configure.): (y_n)
are two Cauchy sequences in the rational, real or complex numbers, then the sum Failed to parse (Missing texvc executable; please see math/README to configure.): (x_n + y_n)
and the product Failed to parse (Missing texvc executable; please see math/README to configure.): (x_n y_n)
are also Cauchy sequences.

Generalizations

Cauchy sequences in topological vector spaces

There is also a concept of Cauchy sequence for a topological vector space Failed to parse (Missing texvc executable; please see math/README to configure.): X

Pick a local base Failed to parse (Missing texvc executable; please see math/README to configure.): B

for Failed to parse (Missing texvc executable; please see math/README to configure.): X
about 0; then (Failed to parse (Missing texvc executable; please see math/README to configure.): x_k

) is a Cauchy sequence if for all members Failed to parse (Missing texvc executable; please see math/README to configure.): V

of Failed to parse (Missing texvc executable; please see math/README to configure.): B

, there is some number Failed to parse (Missing texvc executable; please see math/README to configure.): N

such that whenever 

Failed to parse (Missing texvc executable; please see math/README to configure.): n,m > N, x_n - x_m

is an element of Failed to parse (Missing texvc executable; please see math/README to configure.): V

. If the topology of Failed to parse (Missing texvc executable; please see math/README to configure.): X

is compatible with a translation-invariant metric Failed to parse (Missing texvc executable; please see math/README to configure.): d

, the two definitions agree.

Cauchy sequences in groups

There is also a concept of Cauchy sequence in a group Failed to parse (Missing texvc executable; please see math/README to configure.): G

Let Failed to parse (Missing texvc executable; please see math/README to configure.): H=(H_r)

be a decreasing sequence of normal subgroups of Failed to parse (Missing texvc executable; please see math/README to configure.): G
of finite index.

Then a sequence Failed to parse (Missing texvc executable; please see math/README to configure.): (x_n)

in Failed to parse (Missing texvc executable; please see math/README to configure.): G
is said to be Cauchy (w.r.t. Failed to parse (Missing texvc executable; please see math/README to configure.): H

) if and only if for any Failed to parse (Missing texvc executable; please see math/README to configure.): r

there is Failed to parse (Missing texvc executable; please see math/README to configure.): N
such that Failed to parse (Missing texvc executable; please see math/README to configure.): \forall m,n > N, x_n x_m^{-1} \in H_r

.

The set Failed to parse (Missing texvc executable; please see math/README to configure.): C

of such Cauchy sequences forms a group (for the componentwise product), and the set Failed to parse (Missing texvc executable; please see math/README to configure.): C_0
of null sequences (s.th. Failed to parse (Missing texvc executable; please see math/README to configure.): \forall r, \exists N, \forall n > N, x_n \in H_r

) is a normal subgroup of Failed to parse (Missing texvc executable; please see math/README to configure.): C . The factor group Failed to parse (Missing texvc executable; please see math/README to configure.): C/C_0

is called the completion of Failed to parse (Missing texvc executable; please see math/README to configure.): G
with respect to Failed to parse (Missing texvc executable; please see math/README to configure.): H

.

One can then show that this completion is isomorphic to the inverse limit of the sequence Failed to parse (Missing texvc executable; please see math/README to configure.): (G/H_r) .

An example of this construction, familiar in number theory and algebraic geometry is the construction of the p-adic completion of the integers with respect to a prime p. In this case, G is the integers under addition, and H_r is the additive subgroup consisting of integer multiples of p^r.

If Failed to parse (Missing texvc executable; please see math/README to configure.): H

is a cofinal sequence (i.e., any normal subgroup of finite index contains some Failed to parse (Missing texvc executable; please see math/README to configure.): H_r

), then this completion is canonical in the sense that it is isomorphic to the inverse limit of Failed to parse (Missing texvc executable; please see math/README to configure.): (G/H)_H , where Failed to parse (Missing texvc executable; please see math/README to configure.): H

varies over all normal subgroups of finite index.

For further details, see ch. I.10 in Lang's "Algebra".

In constructive mathematics

In constructive mathematics, Cauchy sequences often must be given with a modulus of Cauchy convergence to be useful. If Failed to parse (Missing texvc executable; please see math/README to configure.): (x_1, x_2, x_3, ...)

is a Cauchy sequence in the set Failed to parse (Missing texvc executable; please see math/README to configure.): X

, then a modulus of Cauchy convergence for the sequence is a function Failed to parse (Missing texvc executable; please see math/README to configure.): \alpha

from the set of natural numbers to itself, such that Failed to parse (Missing texvc executable; please see math/README to configure.): \forall k \forall m, n > \alpha(k), |x_m - x_n| < 1/k

.

Clearly, any sequence with a modulus of Cauchy convergence is a Cauchy sequence. The converse (that every Cauchy sequence has a modulus) follows from the well-ordering property of the natural numbers (let Failed to parse (Missing texvc executable; please see math/README to configure.): \alpha(k)

be the smallest possible Failed to parse (Missing texvc executable; please see math/README to configure.): N
in the definition of Cauchy sequence, taking Failed to parse (Missing texvc executable; please see math/README to configure.): r
to be Failed to parse (Missing texvc executable; please see math/README to configure.): 1/k

). However, this well-ordering property does not hold in constructive mathematics (it is equivalent to the principle of excluded middle). On the other hand, this converse also follows (directly) from the principle of dependent choice (in fact, it will follow from the weaker AC₀₀), which is generally accepted by constructive mathematicians. Thus, moduli of Cauchy convergence are needed directly only by constructive mathematicians who (like Fred Richman) do not wish to use any form of choice.

That said, using a modulus of Cauchy convergence can simplify both definitions and theorems in constructive analysis. Perhaps even more useful are regular Cauchy sequences, sequences with a given modulus of Cauchy convergence (usually Failed to parse (Missing texvc executable; please see math/README to configure.): \alpha(k) = k

or Failed to parse (Missing texvc executable; please see math/README to configure.): \alpha(k) = 2^k

). Any Cauchy sequence with a modulus of Cauchy convergence is equivalent (in the sense used to form the completion of a metric space) to a regular Cauchy sequence; this can be proved without using any form of the axiom of choice. Regular Cauchy sequences were used by Errett Bishop in his Foundations of Constructive Analysis, but they have also been used by Douglas Bridges in a non-constructive textbook (ISBN 978-0-387-98239-7). However, Bridges also works on mathematical constructivism; the concept has not spread far outside of that milieu.

References

• Bourbaki, Nicolas (1972). Commutative Algebra, English translation, Addison-Wesley. ISBN 0-201-00644-8. 
• Lang, Serge (1997). Algebra, 3rd ed., reprint w/ corr., Addison-Wesley. ISBN 978-0-201-55540-0. 
• Spivak, Michael (1994). Calculus, 3rd ed., Berkeley, CA: Publish or Perish. ISBN 0-914098-89-6. 
• Troelstra, A. S.; D. van Dalen. Constructivism in Mathematics: An Introduction.  (for uses in constructive mathematics)ar:متتالية كوشي

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